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\begin{equation*}L = \int_1^3 {\sqrt{1 + \left(x^2 - {1 \over 4x^2}\right)^2}} = \left[ {x^3 \over 3} - {1 \over 4x} \right]_1^3 = \left(9 - {1 \over 12}\right) - \left({1 \over 3} - {1 \over 4}\right) = 8 {5 \over 6}\end{equation*}
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